Pattern inspections for checking masks such as reticles and other photo-masks to be used for manufacturing semiconductor integrated circuits are required to be ever more rigorous to respond to the demand for micronized patterns in recent years.
Generally, pattern inspections involve preparing a real image by irradiating the mask to be used with a laser beam or a charged particle beam, also preparing a reference image by computations, using design data of corresponding areas, and comparing the real image and the reference image to find out discrepancies, if any.
To realize a highly accurate defect inspection, it is necessary that the pattern on the real image and the pattern on the corresponding reference image accurately agree with each other for defect-free areas.
Then, the blur of the image, if any, that is produced when the mask prepared by using design data is observed by way of an optical system has to be reproduced by simulation.
A complex transmittance image produced by a mask is defined as E(x, y) for the following description.
The image intensity distribution obtained by way of an optical system is defined as I(x, y) for simulation.
The optical system may be a coherent coupling system or an incoherent coupling system. The impulse response may be defined as K(x, y) and a convolutional operation may be expressed by * as described, for instance, in Non-Patent Document 1.
Then, the image intensity distribution I(x, y) is expressed by formula (1) below for a coherent coupling system,
[formula 1]I(x, y)=|E*K|2  (1)and by formula (2) below for an incoherent coupling system,[formula 2]I(x, y)=|E|2*K|2  (2)so that it can be determined by a convolutional operation in either case.
The impulse response K is a quantity that can be determined from the pupil function and the wave front aberration of the lens.
As described in Non-Patent Document 2, for instance, the image intensity distribution in a partially coherent coupling system is expressed from the Hopkins theory
                                                        [                              formula                ⁢                                                                  ⁢                3                            ]                        ⁢                                                  ⁢                                          I                ⁡                                  (                                      x                    ,                    y                                    )                                            =                              ∫                                  ∫                                      ∫                                          ∫                                                                        E                          ⁡                                                      (                                                                                          x                                ′                                                            ,                                                              y                                ′                                                                                      )                                                                          ⁢                                                                              E                            *                                                    ⁡                                                      (                                                                                          x                                ~                                                            ,                                                              y                                ~                                                                                      )                                                                          ⁢                                                  J                          ⁡                                                      (                                                                                                                            x                                  ′                                                                -                                                                  x                                  ~                                                                                            ,                                                                                                y                                  ′                                                                -                                                                  y                                  ~                                                                                                                      )                                                                          ⁢                                                  K                          ⁡                                                      (                                                                                          x                                -                                                                  x                                  ′                                                                                            ,                                                              y                                -                                                                  y                                  ′                                                                                                                      )                                                                          ⁢                                                                              K                            *                                                    ⁡                                                      (                                                                                          x                                -                                                                  x                                  ~                                                                                            ,                                                              y                                -                                                                  y                                  ~                                                                                                                      )                                                                          ⁢                                                  ⅆ                                                      x                            ′                                                                          ⁢                                                  ⅆ                                                      y                            ′                                                                          ⁢                                                  ⅆ                                                      x                            ~                                                                          ⁢                                                  ⅆ                                                      y                            ~                                                                                                                                                                            ,                ⁢                                                                  where J is a function that is referred to as mutual intensity function and can be determined from the pupil function of the objective lens and the condenser lens. J=1 in the case of coherent image formation, whereas J=δ(x, y), which is the Derac's delta function, in the case of incoherent image formation.
For example, Patent Document 1 describes a method of computationally determining the image intensity distribution by using Hopkins theory or the M. Yeung's method, which is a practical application of Hopkins theory, for a complex transmittance image showing a mask.
However, Hopkins theory and M. Yeung's method involve computational operations to a large extent and are hence not easy. In other words, they are not suitable for defect inspection apparatus that are required to check defects in a realistic period of time.
Thus, there are attempts to approximate the operation of Hopkins theory by a convolutional operation.
For example, Non-Patent Document 2 describes a technique as summarily shown below.
If the impulse response computationally determined from the pupil function and the wave front aberration is K and p and f are defined respectively by formulas 4 and 5 below,
                              [                      formula            ⁢                                                  ⁢            4                    ]                ⁢                                  ⁢                                  ⁢                              μ            ⁡                          (                                                x                  ~                                ,                                  y                  ~                                            )                                =                                    J              ⁡                              (                                                      x                    ~                                    ,                                      y                    ~                                                  )                                                    H              ⁡                              (                                  0                  ,                  0                                )                                                    ⁢                                  ⁢                                  ⁢        and                            (        3        )                                          [                      formula            ⁢                                                  ⁢            5                    ]                ⁢                                  ⁢                                            f              ⁡                              (                                                      x                    -                                          x                      ′                                                        ,                                      y                    -                                          y                      ′                                                                      )                                      =                                          ∫                                  ∫                                                                                                                                      K                          ⁡                                                      (                                                                                          x                                -                                                                  x                                  ′                                                                -                                                                  x                                  ~                                                                                            ,                                                              y                                -                                                                  y                                  ′                                                                -                                                                  y                                  ~                                                                                                                      )                                                                                                                      2                                        ⁢                                          μ                      ⁡                                              (                                                                              x                            ~                                                    ,                                                      y                            ~                                                                          )                                                              ⁢                                          ⅆ                                              x                        ~                                                              ⁢                                          ⅆ                                              y                        ~                                                                                                                        ∫                                  ∫                                                                                                                                      K                          ⁡                                                      (                                                                                          x                                -                                                                  x                                  ′                                                                -                                                                  x                                  ~                                                                                            ,                                                              y                                -                                                                  y                                  ′                                                                -                                                                  y                                  ~                                                                                                                      )                                                                                                                      2                                        ⁢                                          ⅆ                                              x                        ~                                                              ⁢                                          ⅆ                                              y                        ~                                                                                                                          ,                                    (        4        )            the two point spread functions Kc and Ki are computed in a manner as shown below[formula 6]Kc=f1/2KKi=(1−f)1/2K and the image intensity distribution is computationally determined by the formula shown below,[formula 7]I(x, y)=|E*Kc|2+|E|2*|Kf|2  (5)by using Kc and Ki.
Since J=1 in the case of a coherent image formation system, μ=1 is obtained by using it as substitute in the formula (3). Then, f=1 is obtained by using the latter as substitute in the formula (4). Thus, ultimately the formula (5) agrees with the formula (1). On the other hand, since J=δ(x, y) in the case of an incoherent image formation system, μ=0 is obtained by using it as substitute in the formula (3). Then, f=0 is obtained by using the latter as substitute in the formula (4). Thus, ultimately the formula (5) agrees with the formula (2). In short, the formula (5) can be seen as an approximating expression of a coherent image formation system and a partially coherent image formation system formed by expanding an incoherent image formation system.
Patent Document 2 shows another technique of approximating a partially coherent image formation system with which point spread functions P and Q are computed by using the impulse response K that is also computationally determined from the pupil function and the wave front aberration in a manner as expressed by the formula shown below.
      [          formula      ⁢                          ⁢      8        ]              P      ⁡              (                  x          ,          y                )              =                  J        ⁡                  (                      0            ,            0                    )                    ⁢                                              K            ⁡                          (                              x                ,                y                            )                                                2                        Q      ⁡              (                  x          ,          y                )              =          {                                    0                                                              (                                  x                  ,                  y                                )                            =                              (                                  0                  ,                  0                                )                                                                                        2              ⁢                                                                    K                    *                                    ⁡                                      (                                          0                      ,                      0                                        )                                                  ·                                  J                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ·                                  K                  ⁡                                      (                                          x                      ,                      y                                        )                                                                                            otherwise                              
Then, the image intensity distribution is computationally determined by using P and Q in a manner as expressed by the formula shown below.
[formula 9]I(x, y)=P(x, y)*|E(x, y)|2+Re[E*(x, y){Q(x, y)*E(x, y)}]
However, the partially coherent image formation models shown above do not take optical system errors into consideration.
As described in the Non-Patent Document 3, more complex models such as vector image formation models are being discussed because various optical system errors that cannot be reproduced by a partially coherent model are found.
Patent Document 1: JP-07-220995-A
Patent Document 2: JP-2002-107309-A
Patent Document 3: JP-11-304453-A
Patent Document 4: JP-10-325806-A
Patent Document 5: JP-11-211671-A
Non-Patent Document 1: Tyohiko Yatagai, “Physics of Contemporaries 5: Light and Fourier Transform”, Asakura Shoten, 1992, pp. 92-97.
Non Patent Document 2: Boaz Salik et al., “Average Coherence Approximation for Partially Coherent Optical Systems”, Journal of the Optical Society of America A, 1996, Vol. 13, No. 10, pp. 2086-2090.
Non-Patent Document 3: Tatsuhiko Higashiki, “Optical Lithography Technology—Practical Bases and Problems”, ED Research, Focus Report Series, 2002, pp. 45-49.
Non-Patent Document 4: Hisamoto Hiyoshi, “Expansion of Gravity Center Coordinates Using Voronoi Diagram and Application Thereof to Multi-Dimensional Data Interpolation”, Applied Mathematics, 2002, pp. 176-190.
Non-Patent Document 5: Richard O. Duda et al., “Pattern Classification (2nd ed.)”, translated by Morio Onoe, 2003, pp. 111-113.
Non-Patent Document 6: W. H. Press, P. Flannery, S. A. Teukolsky, W. T. Vetterling, “Numerical Recipes”, translated by Katsuichi Tankei, Haruhiko Okumura, Toshiro Sato, Makoto Kobayashi, Gijutsu-Hyoron Co., Ltd, published on Nov. 1, 2001, pp. 307-312.